FamaFrench and Merck Gamma

Fama-French 3-factor model: purpose

Explains a stock's (or portfolio's) expected excess return. CAPM has one factor (the market); Fama and French found two more risk factors, one for size and one for value.

Fama-French expected return form

E(R) = Rf + b1·E(MKT) + b2·E(SMB) + b3·E(HML)

Fama-French regression equation (for testing)

Ri - Rf = a + b1(MKT) + b2(SMB) + b3(HML) + e

Ri - Rf (dependent variable)

The stock's excess return: its return minus the risk-free rate. This is the 'Y' in the regression.

Rf

The risk-free rate, such as the 3-month T-bill rate, converted to the same period as the returns (e.g., daily). Subtracted from returns to get excess returns.

a (intercept / alpha)

The part of the excess return NOT explained by the three factors. The model implies the intercept is zero, so it is the key thing you test.

MKT factor and b1

MKT = market return minus the risk-free rate (the market risk premium). b1 is the stock's sensitivity to the market, like beta in CAPM.

SMB (Small Minus Big)

The size factor (firm-size premium): return on small-cap stocks minus return on large-cap stocks. Captures the tendency of small firms to earn higher returns.

b2 (SMB loading)

Sensitivity of the stock to the size factor. Positive means it behaves like a small-cap stock; negative means it behaves like a large-cap stock.

HML (High Minus Low)

The value factor (value premium): return on high book-to-market (value) stocks minus return on low book-to-market (growth) stocks. Fama treats high book-to-market as a proxy for default risk.

b3 (HML loading)

Sensitivity of the stock to the value factor. Positive means it behaves like a value stock; negative means it behaves like a growth stock.

e (error term)

The random, firm-specific part of the return that the factors do not explain.

Three factors of Fama-French

1) Market factor (Rm - Rf), 2) Size factor (SMB), 3) Value factor (HML).

How Fama-French differs from CAPM

CAPM uses only one factor (market risk/beta). Fama-French adds SMB (size) and HML (value) to explain returns that CAPM leaves unexplained.

Data needed to test Fama-French

The stock or portfolio returns, the risk-free rate (e.g., 3-month T-bill converted to the same period), and the MKT, SMB and HML factor data from the Fama-French data library, all for the same periods.

Step 1 of testing the model

Get the stock or portfolio returns, the risk-free rate, and the MKT, SMB and HML factor data from the Fama-French data library for the same periods.

Step 2 of testing the model

Compute the stock's excess return each period: stock return minus the risk-free rate.

Step 3 of testing the model

Regress the excess return on the factors: Ri - Rf = a + b1(MKT) + b2(SMB) + b3(HML). Excess return is Y; MKT, SMB and HML are the X variables.

Step 4 of testing the model

State the hypothesis: the model implies the intercept is zero, so test H0: a = 0.

Step 5 of testing the model

Check whether the intercept is statistically close to zero (|t| < 2 or a high p-value). For a set of portfolios, check whether the average intercept is close to zero.

Null hypothesis when testing Fama-French

H0: a = 0. If the model works, the three factors explain all of the expected excess return, leaving no intercept.

Decision rule for accepting the model

Look at the intercept's t-stat. If |t| < 2 (or p-value is high), the intercept is statistically zero: cannot reject H0, so ACCEPT the model. If |t| > 2, reject H0: the model is missing something that explains returns.

What the accept/reject decision is based on

The INTERCEPT only. Not on whether the factor coefficients are significant and not on R Square.

Insignificant SMB or HML for a stock

Means that factor does not individually explain this stock's returns. It does NOT affect the accept/reject decision, which rests on the intercept.

Reading a factor coefficient

The coefficient is the stock's sensitivity to that factor (e.g., a MKT coefficient above 1 means the stock moves more than the market risk premium). |t| above 2 or a low p-value means it is significant.

t-stat

Coefficient divided by its standard error. Measures how many standard errors the estimate is away from zero; roughly |t| of 2 or more is significant.

p-value

The probability of getting a coefficient this far from zero if the true value were zero. Small p-value (below 0.05) = statistically significant.

R Square

The share of the variation in the stock's excess return that the three factors explain together. Report it, but do not base the accept/reject decision on it.

Adjusted R Square

R Square adjusted for the number of explanatory variables; penalizes adding factors that do not help.

Significance F

Tests whether the factors jointly explain the returns. A small Significance F means the regression as a whole is significant (at least one factor matters).

Low R Square but significant F

The factors as a group do help explain returns, but they explain only a small part of the total variation; much of the stock's risk is idiosyncratic.

Project Gamma deal (setup)

Merck would make payments to Gamma (a small biotech company) over several years, owe royalties if the product reached market, and could terminate the agreement at any time.

Real option

The right, but not the obligation, to take a business action (invest, expand, delay, abandon) in the future. Valued like a financial option.

Why an option approach fit Project Gamma better than NPV

Cash flows and market conditions were extremely hard to predict (development takes up to a decade), so NPV may miss the project's strategic value. Returns were asymmetric: large upside, downside limited to the initial investment, and Merck had the right but not the obligation to keep investing.

Asymmetric returns (Project Gamma)

Large potential upside, but the downside is limited to the initial investment because Merck can terminate at any time. This payoff shape is what makes it an option.

Problem with traditional NPV for R&D

NPV treats the investment as a fixed commitment and ignores management's flexibility to wait, learn, and abandon, so it can understate the project's real value.

How uncertainty affects option value

Higher uncertainty (volatility) INCREASES option value, because the downside is limited (you can walk away) while the upside is kept.

Merck's up-front payment to Gamma = ?

The price (premium) Merck pays to buy the option.

Black-Scholes five inputs

1) Exercise price, 2) Stock price (value of the underlying asset), 3) Time to expiration, 4) Volatility, 5) Risk-free interest rate.

Exercise price (Project Gamma)

The capital investment Merck would make about two years later if it decided to go ahead with the project.

Stock price / underlying asset (Project Gamma)

The present value of the project's cash flows, EXCLUDING the later capital investment and the PV of the up-front fees and development costs over the next two years.

Time to expiration (Project Gamma)

Merck varied it over two, three and four years. Exercisable in two years at the earliest; structured to expire in four years because Merck expected competing products by then, making market entry unfeasible.

Earliest exercise vs. expiration (Project Gamma)

Two years is the EARLIEST exercise date; the option was structured to EXPIRE at four years (per the article). Mention the range on the exam.

Volatility (Project Gamma)

Not observable for the project, so Merck used the annual standard deviation of returns for typical biotechnology stocks (from an investment bank) as a proxy, using a conservative range of 40% to 60%.

Risk-free interest rate (Project Gamma)

Assumed 4.5%, roughly the U.S. Treasury rate over the two-to-four-year period to expiration.

Merck's conclusion on Project Gamma

The option's value was significantly more than the up-front payment that needed to be invested, so the deal was worth doing.

Call option analogy for Project Gamma

Merck pays a premium (up-front payment) for the right to later 'buy' the project's cash flows (underlying asset) by paying the capital investment (exercise price).

Option to abandon

Merck's ability to terminate the agreement at any time. It limits losses to what has already been paid and adds value to the deal.

How time to expiration affects option value

A LONGER time to expiration INCREASES option value: more time for the project's value to move up while the downside stays limited. Like higher volatility, it makes the option worth more.

Decision rule: option value vs. up-front payment

If the option's value is greater than the up-front payment (the premium), do the deal. If it is less, the option costs more than it is worth.

NPV including option value

Total value = traditional (static) NPV + value of the real option. A project with a negative static NPV can still be worth doing if the option value is large enough.

Number of distinct terms in a variance-covariance box (N assets)

N(N+1)/2 total: N variances (on the diagonal) plus N(N-1)/2 covariances (each covariance appears twice, mirrored across the diagonal).

Why the variance-covariance box is symmetric

Cov(1,2) = Cov(2,1), so each covariance appears twice, mirrored across the diagonal. Only the diagonal holds variances.

Negative portfolio weight

Means you short-sell that asset: you borrow and sell it, and use the cash to buy more of the other assets. Only possible if short selling is allowed for that asset.

No short selling constraint

Weight >= 0 for each asset that cannot be shorted. Assets without this constraint can end up with negative weights.